$\begin{aligned} \vec{a} &=\hat{i}-2 \hat{j}+5 \hat{k} \\ \vec{b} &=2 \hat{i}+\hat{j}-3 \hat{k} \\ & \vec{b}-\vec{a}=2 \hat{i}+\hat{j}-3 \hat{k}-(\hat{i}-2 \hat{j}+5 \hat{k}) \\ &=\hat{i}(2-1)+\hat{j}(1+2)+\hat{k}(-3-5)=\hat{i}+3 \hat{j}-8 \hat{k} \\ & 3 \vec{a}+\vec{b}=3(\hat{i}-2 \hat{j}+5 \hat{k})+(2 \hat{i}+\hat{j}-3 \hat{k}) \\ &=3 \hat{i}-6 \hat{j}+15 \hat{k}+2 \hat{i}+\hat{j}-3 \hat{k} \\ &=\hat{i}(3+2)+\hat{j}(-6+1)+\hat{k}(15-3) \\ &=5 \hat{i}-5 \hat{j}+12 \hat{k} \\ &(\vec{b}-\vec{a}) \cdot(3 \vec{a}+\vec{b})=(\hat{i}+3 \hat{j}-8 \hat{k}) \cdot(5 \hat{i}-5 \hat{j}+12 \hat{k}) \\ &=(1)(5)+(3)(-5)+(-8)(12) \\ &=5-15-96=-106 \end{aligned}$